UNITED NATIONS, Aug 10 (Reuters) - The man tipped to replace Kofi Annan as the U.N.-Arab League mediator on Syria, Lakhdar Brahimi of Algeria, urged world leaders on Friday to overcome their differences on the 17-month-old conflict that is slipping deeper into full-scale civil war.

“The U.N. Security Council and regional states must unite to ensure that a political transition can take place as soon as possible,” Brahimi said in a statement published on the website of The Elders, an independent group of global leaders committed to peace and human rights.

“Millions of Syrians are clamoring for peace,” Brahimi said. “World leaders cannot remain divided any longer, over and above their cries.”

It is Brahimi’s first public statement on Syria since diplomats told Reuters on Thursday that U.N. Secretary-General Ban Ki-moon was expected to name the veteran Algerian diplomat as early as next week to replace Annan, barring a last-minute change or cold feet on Brahimi’s part.

U.N. diplomats said Brahimi has told Ban and Arab League chief Nabil Elaraby that his condition for accepting the job was that he receive “strong support” from the Security Council, which has been sharply divided on Syria for since the uprising began in March 2011.

It was not immediately clear what Brahimi meant by “strong support,” though the diplomats said he was understandably reluctant to take a job that would be extremely difficult to succeed at.

Brahimi, 78, has served as a U.N. special envoy in a series of challenging circumstances, including in Iraq after the U.S. invasion that toppled Saddam Hussein; in Afghanistan both before and after the end of Taliban rule, and in South Africa as it emerged from the apartheid era.

French U.N. Ambassador Gerard Araud, president of the Security Council this month, told reporters Ban was expected to make an announcement about Annan’s successor on Monday or Tuesday after consulting with Elaraby.

Annan, a former U.N. secretary-general and Nobel Peace Prize laureate, said he would step down on Aug. 31 because he was not able to carry out his job with the U.N. Security Council’s veto powers hopelessly divided and deadlocked.

There are no signs that Brahimi will get his wish anytime soon, if at all. The divisions on the Security Council - above all the split between the United States and Russia - run deep.

Russia, with the aid of China, has vetoed three resolutions criticizing and threatening sanctions against Damascus for its attempt to use military force and heavy arms to crush an increasingly militant opposition.

Washington, U.N. diplomats said, saw little point in replacing Annan since Moscow continues to support Syrian President Bashar al-Assad and opposes sanctions intended to pressure Damascus into halting the violence.